Prove that the 99th power of 2 plus the 99th power of 3 can be divided by 5

Prove that the 99th power of 2 plus the 99th power of 3 can be divided by 5

The last digit of the fourth power of 2 is 6
The last digit of the 99th power of 2
=The 24th power of (the 4th power of 2) × The last digit of the cube of 2
=6 × Last digit of 8
=8
Similarly:
The last digit of the 99th power of 3
=(4th power of 3) 24th power of × three ³ Last digit of
=1 × Last digit of 27
=7
The 99th power of 2 plus the last digit of the 99th power of 3
=8 + 7 last digit
=5
So it can be divided by five

Verify that the 55th power of 55 + 9 can be divided by 8, .

Just consider that 55 ^ 55 + 1 can be divided by 8. Because 55 ^ 55 + 9 = 55 ^ 55 + 9 = [(56-1) ^ 55 + 1] + 8, and 55 ^ 55 + 1 = (56-1) ^ 55 + 1. Here, the first 55 terms of the (56-1) ^ 55 expansion contain a factor 56, so they are multiples of 8. And the 56th term is - 1, which offsets the + 1 in (56-1) ^ 55 + 1, so 55 ^ 55 + 1 can be divided by 8. And 8 is also a multiple of itself. Therefore, 55 ^ 55 + 9 can be divided by 8

It is proved that the 10th power of polynomial 7 - the 9th power of polynomial 7 - the 8th power of polynomial 7 can be divided by 41

A common factor 7 ^ 7 can be proposed
Namely
7^10 - 7^9 - 7^8
= 7^8 × ( 7^2 - 7 - 1 )
= 7^8 × ( 49 - 7 - 1 )
= 7^8 × forty-one
That is, the original polynomial can be divided by 41

It is proved that the 10th power of 2 - the 8th power of 2 + the 6th power of 2 - the 4th power of 2 + the 2nd power of 2 - 1 can be divided by 9 Speak clearly and write clearly

(2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 1, 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 1 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^ 2 ^

Calculate the 2nd power of 50 - the 2nd power of 49 + the 2nd power of 48 - the 2nd power of 47 +... + the 2nd power of 2 - 1

Pairing in pairs, using the square difference formula. (50 ^ 2-49 ^ 2) + (48 ^ 2-47 ^ 2) + (46 ^ 2-45 ^ 2)... (4 ^ 2-3 ^ 2) + (2 ^ 2-1 ^ 2) = 50 + 49 + 48 + 47 +... 4 + 3 + 2 + 1 = 25 * 51 = 1275

The 51st power of negative 2 + the 50th power of negative 2 = how much (process) ～

The 51st power of negative 2 + the 50th power of negative 2
=-2 * 2 to the 50th power + 2 to the 50th power
=-The 50th power of 2
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